Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ C2×#97 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | AGL(1,𝔽5)2 |
| commutator, socle | C2×C52 | C2×C42 |
| Frattini | C2 | C2×AGL(1,𝔽5)2 |
| Fitting | C4×C52 | C42 |
| prime radical | C2×D52 | C2×C4 |
| cosocle | D5×Dic5 | C23 |
| hypocenter | C52 | C2×#97 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C4 | |
303 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^6,5^2|a^2=d,b^2=e,c^2=f,e^2=f,b^a=bf,h^a=h^2,g^b=g^2,h^d=h^4,g^e=g^476 conjugacy classes — view the full character table.
| maximal subgroup of | #72242321 |
| maximal normal subgroup of | #72242321 |
| maximal quotient of | #72242309 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9804);